# B7.3 Further Quantum Theory - Material for the year 2020-2021

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Part A Quantum Theory.

16 lectures

### Assessment type:

- Written Examination

This course builds on Part A Quantum Theory. The mathematical foundations of quantum theory are developed more deeply than in the Part A course, and general principles regarding the realization of symmetries in quantum mechanical systems are developed. Systems of several particles are studied, including a consideration of identical particles and particle statistics. Along the way, several simple-but-relevant concepts from the theories of Lie groups, representation theory, and functional analysis are introduced in a hands-on fashion.

In semi-realistic systems, an exact solution to the quantum dynamics is rarely forthcoming. A number of important approximation techniques are also developed. These are employed to address problems such as the determination of energy levels of the Helium atom.

Student will be able to state the postulates of nonrelativistic quantum theory and explain how they are realized in key examples. They will be able to analyse simple quantum systems that admit exact solutions by exploiting symmetries and algebraic techniques. They will be able to calculate approximations to energy-levels and other properties of more complicated systems using perturbation theory, semi-classical techniques, and variational methods.

Abstract formulation of quantum mechanics in terms of linear operators on Hilbert spaces; Dirac notation; discrete and continuum states; time evolution and the propagator.

Systems of several particles and Hilbert space tensor products; distinguishable and indistinguishable particles; Fermi-Dirac and Bose-Einstein statistics; Pauli exclusion principle; elementary aspects of quantum entanglement.

Symmetries in quantum mechanics as unitary and anti-unitary operators; rotations, angular momentum, and spin; spin-1/2 and projective representations of SO(3); addition of angular momentum; Spin-statistics theorem.

Approximation methods: Rayleigh-SchrÃ¶dinger perturbation theory; variational methods; WKB approximation and Bohr-Sommerfeld quantization.

Time-dependence in the Heisenberg, SchrÃ¶dinger, and interaction/Dyson; time-dependent perturbation theory and the Feynman-Dyson expansion.

Elementary scattering theory.

- Steven Weinberg,
*Lectures on Quantum Mechanics*(CUP, 2015). - J. J. Sakurai and Jim Napolitano,
*Modern Quantum Mechanics*(CUP, 2017). - Keith Hannabuss,
*An Introduction to Quantum Theory (OUP, 1997).*

*Further Reading:*

- David Griffiths and Darrell Schroeter,
*Introduction to Quantum Mechanics*(CUP, 2017). - B. Hall,
*Quantum Theory for Mathematicians*(Springer, 2013). - Eugene Merzebcher,
*Quantum Mechanics*(Wiley, 1970).